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PLAN OF AMPHIPOLIS TOMB
By Prof. L. Kaliambos (Natural Philosopher in New Energy) December 11, 2015 (Σχέδιο του Τάφου της Αμφίπολης) INTRODUCTION In August of 2014 the excavation team of Amphipolis announced incorrectly that the diameter D of the cone pyramid of Amphipolis is D = 158.4 m. as a result of the perimeter (p = 497 m) measured outside the surrounding circular wall. In fact, the correct perimeter of the cone pyramid is π = 3.1416Χ157.5 m = 494.8 m because Dinocrates used not only the correct number π = 3.1416 but also the diameter d = 1 Hellenistic stadium = 157.5 m.This photo is from the interview I gave to the author of the Spiritual Thessaly Mrs Dimitra Bardani about the plan of Amphipolis Tomb made by the architect Dinocrates. Indeed, according to the history of Greek people ( Volume Δ page 208 ) after the death of Hephaestion (324 BC) Alexander the Great ordered his architect Dinocrates for planning an expensive tomb called PYRE for hero Hephaestion having a base of one Hellenistic stadium used also by Eratosthenes who found the circumference of our Earth (see Eratosthenes-WIKIPEDIA). Nevertheless the archeologist K. Peristeri suggested that the tomb was constructed by the famous architect Dinocrates. Note that she was based on the fallacious hypotheses of the architect of the excavation team who tried to find a relationship between the walls of Alexandria and the diameter D of the tomb by suggesting arbitrarily that the Hellenistic stadium is equal to 165 m. Under such false ideas which are in conflict with the historical background the professor, Olga Palagia, of Classical Archaeology in the University of Athens, (Sept 9, 2014) in a criticism wrote that the Amphipolis tomb was constructed not by Dinocrates but by Romans. To avoid such a confusion in November 2014 I published my paper “TOMB OF HEPHAESTION IN AMPHIPOLIS”. Despite the various speculations about the dimensions of the Amphipolis tomb my discovery of the Alexandrian stadion (d = 1 stadion = 157.5 m) of the circular base of the Kasta hill reveals all the secrets of the monument, which give us the sacred numbers like 7, 12 and 3 used in the ancient astronomy. Surprisingly I discovered also that Dinocrates used earlier the same numbers 7, 12, and 3 for planning the perimeter P of the walls of the ancient Alexandria in Egypt (331 BC), since I found that P = (31.5 + 10.5) = 7X12 = 84 stadia, while 31.5/10.5 = 3. Indeed, Alexander the Great after the death of Hephaestion (324 BC) ordered his architect Dinocrates for planning in Babylon the so-called PYRE. But since the PYRE was very expensive after the death of Alexander (323 BC) Antipater in Macedonia (Amphipolis) ordered the architect Dinocrates (320 BC) for transforming the mound of Kasta into a mathematical cone pyramid including the same astronomical numbers of Alexandria. ( See my MATHEMATICAL TOMB OF HERO HEPHAESTION). TRANSFORMATION OF THE KASTA MOUND INTO A MATHEMATICAL CONE PYRAMID FOR HERO HEPHAESTION IN AMPHIPOLIS Here I emphasize that Dinocrates for defining the circumference C = 2πr of the Amphipolis cone pyramid used the radius r = d/2 = 1/2 stadia = 157.5/2 = 78.75 m. However to transform the mound Kasta into a mathematical tomb of height H = 1/7 stadia he should use the slant height (L) by using the Pythagorean theorem as L = (1/22 + 1/72)0.5 = + 72)/(22X72)0.5 = (53/196)0.5 = 0.52 stadia Then starting from the top of the hill (position of the lion) and using the negative slope - H/r = - (1/7)/(1/2) = - 2/7 he determined the circumference 2πr = 494.8 m. On the other hand using the height H = d/7 and the diameter d = 1 stadium he calculated the volume V of the cone pyramid as V = (1/3) h (π/4) d2 = (1/3)(d/7)(π/4)d2 πd3/(7X12) and since d = 1 stadium he got V = π/(7Χ12) cubic stadia In other words for the calculation of the volume of the cone pyramid he used not only the astronomical numbers 7 and 12 as those of Alexandria but also the mathematical constant π = 3.1416 . PLAN OF THE TOTAL HIGHT ( Y = H + h ) OF THE MONUMENT The total height Y of the monument is Y = H + h Here H = d/7 and h is the total height of the lion including its base and its foundation base. Unfortunately the architect M. Lefantzis in order to provide an harmonic relation between the total height (h ) of the lion of Amphipolis and the wrong diameter (D =158.4 m) of the circular base of the Kasta hill increased arbitrarily the total height from the correct h = 13.125 m to the wrong h =15.84 m, so that the ratio being 10 times smaller than the diameter (D = 158.4 m). In fact, according to the “Kasta tomb-Wikipedia” the height of the lion with its base is 8 m. Thus the total height including the foundation base is h = 8 + 5 = 13 m or indetail h = 13.125 m. That is 13.125/157.5 = 1/12 stadia . So to avoid the confusion presented by the excavation team about the total height of the Hephaestion monument I discovered also that Dinocrates using the same astronomical numbers 7 and 12 could determine the total height ( Y ) of the monument. Since the height H of the cone pyramid is equal to d/7 and the height h of the statue of the lion with the two bases is equal to d/12 I discovered that Dinocrates for determining Y used the following math as Y = d/7 + d/12 = d(7+12)/(7X12) Since d = 1 stadium he could also write Y = (7+12)/(7X12) stadia. Since one Alexandrian stadium = 157.5 m , today one gets a total height of the monument: Y = 35.6 m. HOW DINOCRATES USED THE ASTRONOMICAL NUMBER 3 FOR CALCULATING THE VOLUME OF THE SURROUNDING WALL My discovery helps the study of the Hellenistic period, because the mathematical tomb of the Hero Hephaestion in Amphipolis is the only one survived monument which gives us today the unit of length used by Dinocrates. Since Dinocrates worked for the government, no one could order him for personal purposes. It is of interest to note that the d = 1 stadium = 157.5 m is the diameter corresponding to the medium line (mean perimeter) of the surrounding wall along the circular base having a width w = D-d. Surprisingly I discovered that Dinocrates in his plan determined also the second greater diameter D = 158.4 m of a perimeter measured outside the surrounding wall by using the astronomical number 3. In his plan of astronomical numbers for calculating the volume (v) of the marbles of the circular wall he suggested that v = 0.3/103 cubic stadia, which includes the mystic numbers related to 3 = (7X12)/28. As in the case of the walls of Alexandria Dinocrates also suggested that the width (w)of the wall and the height 3w of the same wall should be related with the astronomical number 3 . Here P = π is the length of a parallelepiped in which the height is equal to 3w . So the volume ( v ) of the parallelepiped ( volume of marbles of the surrounding wall ) should be given by v = π(3w)w = 0.3/103 cubic stadia So w2 = 0.3/3π103 and w = D-d = (0.3/3π103)0.5 stadia That is D = d + w = 1 + ( 0.3/3π103)0.5 stadia Since 1 stadium is equal to 157.5 m one gets D = 157.5 + 0.9 = 158.4 m. ' ' DINOCRATES ALSO USED THE GOLDEN SECTION IN AMPHIPOLIS The excavation team had so far unearthed two female statues of the Caryatid type in the antechamber, which support the entrance to the second compartment of the tomb. The height α of each Caryatid is α = 2.27 m. The Caryatids are on a pedestal of height β = 1.40 m , making the total height (α + β) = 3.67 m of the statues. However in the absence of a detailed knowledge about the math and the architecture of ancient Greeks the architect of the excavation team in Amphipolis did not relate such very important numbers with the so-called GOLDEN SECTION used in the Temple of Parthenon and in other ancient monuments. After a careful analysis of such dimensions I discovered that Dinocrates used also the so-called golden ratio or golden section. In mathematics GOLDEN SECTION is the division of a line segment into extreme and mean ratio. This is obtained by dividing a line into two parts a and b such that the square of the one part is equal to the product of the whole segment and the other part. That is (α+β)/α = α/β = φ = (1+50.5)/2 = 1.61803… Or α2 = ( α +β )β Ancient Greek mathematicians found that φ = ( 1 + 50.5)/2 by using a rectangle of sides β and β/2. In this case they found that α = β/2 + x where x is the diagonal of the rectangle. Here x is given by using the Pythagorean Theorem as x2 = β2 + β2/4 = 4β2/4 + β2/4 = 5β2/4 Or x = β(50.5/2). So α = β/2 + x = β/2 + β(50.5/2 ).That is φ = α/β = [ β/2 + β( 50.5/2)] / β = (1 + 50.5)/2 = 1.61803.... An approximate value for the φ is 1.618. Thus Dinocrates starting from the total height (α+β) = 3.67 m was able to find the heights α and β as (α +β)/α = φ Or α = 3.67/1.618 = 2.268 m = 2.27 m and β = 3.67- 2.27 = 1.4 m Note that the Egyptians may have used both π and φ in the design of the Great Pyramids. The Greeks are thought by some to have based the design of the Parthenon on this proportion, but this is subject to some conjecture. Phidias (500 BC – 432 BC), a Greek sculptor and mathematician, studied φ and applied it to the design of sculptures for the Parthenon. Plato (circa 428 BC – 347 BC), in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos. In nature this golden section as a principle may be observed in the arrangements of leaves on a twig, petals on a flower and the arms of the starfish. The ancient Greeks considered a rectangle whose sides are in this ratio to be aesthetically the most pleasing of all rectangles and constructed their buildings on this principle. Category:Fundamental physics concepts